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N94- 17338 


Diffractive Optical Elements for Generating Arbitrary Line Foci 

Joseph N. Mait, Dennis W. Prather, Joseph van der Gracht, and Tristan J. Tayag 

U.S. Army Research Laboratory 
AMSRL-SS-SF 
2800 Powder Mill Road 
Adelphi, Maryland 20783 


1. Introduction 


The key optical component in the architecture of the linearly variable magnification telescope shown 
in Fig. 1 is a conical lens. This architecture has application to Doppler radar processing and to wavelet 
processing. Unfortunately, the unique surface profile of a conical lens does not allow traditional grinding 
techniques to be used for fabrication and therefore its fabrication is considered custom. In addition to the 
requirement of custom fabrication, a refractive conical lens introduces phase aberrations that are intrinsic 
to its conic shape. Further, due to the large prismatic component of the lens, the variable magnification 
telescope architecture is off-axis. 

To overcome the fabrication and application difficulties of a refractive lens, we consider the 
construction of a hybrid diffractive-refractive lens that has a phase profile given by 


<t>{ u * v ) = y [/(«) - v7 2 (») + * 2 ]. 


(i) 


where the focal length f(v) is an arbitrary function of v and A is the wavelength of illumination. The effect 
of such an arbitrary line foci element (ALFE) is to produce a focal line in three-dimensions whose shape 
along the axis of propagation follows that of the function /(v). Cylindrical and conical lenses are the most 
common refractive elements that are capable of producing focal lines. 

To generate the focal line f(v ) we consider a hybrid element that consists of a refractive element that 
generates the focal line /r{v) and a diffractive lens that generates the focal line Jd{ v )' 


+ 


fn{v) }d{v) f(v) 


( 2 ) 


If the refractive element is used to provide a constant optical power, fn{v) — //?, i.e., the refractive element 
is a cylindrical lens, constraints on the fabrication of the diffractive lens can be reduced. For example, an 
ALFE that varies in focal length from 160 mm to 240 mm over a 50 x 50 mm 2 aperture varies in /-number 
from 3.2 to 4.8. If//? — 325 mm, the cylindrical lens is //6.5 and the diffractive lens varies in /-number 
from 6.3 [/( 0) = 315 mm] to 18.35 [/(50) = 917.5 mm], which is less difficult to fabricate than lower 
/-number lenses. 

We now consider the design of a thin phase-only diffractive optical element (DOE) P(u, t>), 

P(tt,v) = exp[/<£i>(u,v)], (3a) 

that is capable of generating the focal line /d{v). A multi-level quantized phase-only DOE is assumed, 


TV M 


<j> D {u,v) - EE 0nm, l rect( 


u — u n v — v„ 


■). 


n = 0 m=0 

where A is the minimum feature size of the pattern generator used to produce the binary masks and 

, , 2tt 27t ,w 

4>nm,L £ [®> 2 ? ' ^ — 


(3b) 


(3c) 


99 

PHBCBDm« PAGE BLANK NOT FILIW 


Conf. on Binary Optics, 1993 


The resolution with which each phase pixel can be positioned is defined by the placement accuracy e of the 
pattern generator, i.e., u n le and v n = fee. The ability of P{u y v ) to generate a high-fidelity focal line 
fv(v) is limited by the minimum feature size A of the pattern generator, its placement accuracy e, and the 
number of phase quantization levels 2 h . Design of Mir diffractive ALFE entails the determination of the 
parameters <f> n m,L and from which h binary masks are produced. 

We consider the design of a diffractive ALFE based on the sampling and quantization of <£p(u, r) 
and experimental results from a binary-phase ALFE designed in this manner are presented. We also 
consider two alternative design approaches: a second approach based on sampling and quantization and an 
iterative approach that deternines the parameters by optimizing some design metric. The difficulties 
encountered in lens design and characterization of lens performance are addressed. 


2. Deterministic Design 

In the literature, the design of diffractive lenses implies the design of spherical lenses, which, by 
exploiting spherical symmetry, can be reduced from a two-dimensional problem to one that is 
one-dimensional. For diffractive ALFE design, symmetry properties can not be exploited to reduce the 
dimensionality of the design problem. Rather, dimensionality reduction is realized by noting that the phase 
function at every u-axis slice of the diffractive ALFE is a spatially scaled version of a one-dimensional lens: 


4>d{u, v) = s(v) ^Dt-y-r.O], 


(4a) 


where «(v) represents a normalized focal function, 


#(i>) = 


fp( v ) 

Mo)' 


(4b) 


The even nature of the ALFE, i.e., <£d(u, v) — can be used to halve the number of design 

parameters. 

Lens design is realized by quantizing lines of constant phase: 


where 


u-tO) = 



2tt 


2 /d(v) 


*i = f?. 


(5a) 

(5b) 


The number of phase lines I to be quantized is dependent upon the minimum phase value obtained by 
<p D (u } v) : < 


7 = ROUND 


‘2 l 

— xnin^fu, v)} 


(5c) 


where ROUND implies rounded to the nearest integer. The minimum phase is, in turn, related to the 
minimum /-number one-dimensional lens described by <fip(u y v). If the DOE aperture is allowed to vary 
with f{v ) then the diffractive ALFE maintains a constant /-number and each lens has the same number of 
phase lines. However, to allow the aperture to vary represents an inefficient use of the overall aperture. On 
the other hand, a constant aperture generates variable /-number one-dimensional lenses and prod uces a 
focal line that has variable spot size and brightness. The tradeoff between aperture efficiency and focal line 
fidelity needs to be consider for each application. We assume that the aperture is constant and is defined 
by the aperture of the refractive cylindrical lens. 

One method for implementing the design, referred to as the direct sampling (DS) method, is to 
ignore the effect of e and quantize the lines u,*(r) according to the minimum feature size A. For the DS 
method, L and A® dictate the minimum /-number of the lens that can be designed and L dictates the fens 


f 


100 


i ii ii i miiMi 1 1 1 n i i i i III mil I anil lull mi ii II I mi || mu 



diffraction efficiency [1,2]. In effect, the DS method assumes that <£p(u, r) is sampled on a regular 
Cartesian grid and is then quantized. The spacing of the grid is determined by the minimum feature size of 
the pattern generator used to produce the binary masks. 

Figure 2(a) is the DS-generated binary mask for a conical lens that has a linear change in focal 
length from 300 mm to 480 mm over a 4 x 4 mm 2 aperture. The focal length increases in 3.6 mm 
increments over the aperture, thus, the ALFE consists of 50 one-dimensional lenses. The minimum feature 
size is 10 /mi. The lens was fabricated on a Corning glass substrate that has an index of refraction n = 1.53 
using a 1.8 /mi layer of positive photoresist (Iloechst AZ 5214) and an Ar ion etch. The axial performance 
of the lens is represented in Figs. 2(b)-(d), which indicates that the focal length actually changes from 
approximately 220 mm to 350 mm. 

Due to the varying axial behavior of the lens, it is difficult to characterize its performance. For a 
spherical lens, in most instances, it suffices to take a measure of the diffraction efficiency, the percentage of 
input light energy that is brought to focus. For an ALFE, the diffraction efficiency must be measured along 
the focal line: 

v{y) = J \U[x,y, f(y )]\ 2 dx/ J \U[x,y\ /(y)]| 2 dx (6) 

where X is the width of the focal spot and U(z, y\ z) is the complex wave-amplitude field generated by 
P(u, v) at an arbitrary distance z in the near-field, 

U{x,y\z) - exp j exp [j<j> D (u, v)] exp - *) 2 + (t> - y) 2 ]) dudv. (7) 

The area of the ALFE aperture is denoted by A. To reduce the computational load in evaluating the 
three-dimensional function U(x i y; z), if y-axis diffraction is ignored, Eq. (7) can be simplified 


U[x,y;f{y)\ = — exp 
3 X 


2 ?r/(y)' 


/ 

Jw u 


exp[j</>i>(M,y)] exp 


A/(y) 


{u-xf 


du, 


( 8 ) 


where IT r u is the u-axis aperture of the ALFE. The diffraction efficiency along the focal line generated by 
the DS conical lens as determined using Eq. (8) is presented in Fig. 3. 

Diffraction efficiency can be increased if the placement accuracy e is not ignored. In other words, 
although the minimum feature size is fixed, the accuracy with which the lines of constant phase are 
quantized can be sampled on a finer grid determined by the placement accuracy. Fabrication of the masks, 
though, is still limited by the minimum feature size of the pattern generator. As represented in Fig. 4, to 
realize this finer quantization multiple exposures are used to generate the masks necessary for fabrication, 
but, because the masks are binary, multiple exposures do not affect mask transmission. This design 
technique is referred to as analytic quantization (AQ). 

The distinction between DS- and AQ-designed ALFEs is evident in Fig. 5 and the increase in 
diffraction efficiency is represented in Fig. 6. The ALFE is characterized by a focal length change from 256 
//m to 1040 /xrn in 19.6 /xm increments (40 lenses) over a 80 x 80 /xm 2 aperture. Minimum feature size is 
assumed to be 2 /xm with a placement accuracy of 0.2 fim. The percent change in diffraction efficiency 
ranges from a minimum of zero to a high of 35% and is most notable for low /-number lenses, which have 
higher phase curvature than high /-number lenses. Note that the improvement in diffraction efficiency is 
reduced for those lenses that experience “edge effects” introduced by the minimum feature size limitation. 
Unfortunately", consideration of the placement accuracy during design increases by 100 the amount of data 
necessary for fabrication of a single mask. Thus, whereas the lens in Fig. 5(a) contains only 40 x 40 data 
points, the lens in Fig. 5(b) contains 400 x 400. 


3. Iterative Design 

Although performance characterization of an ALFE requires, in general, the evaluation of Eq. (7), 
the application to a variable magnification telescope allows us to characterize lens performance based on 


101 


telescope performance. For telescopic operation, the input and output are collimated, i.e., the input object 
is located at — oo and the output image is at -foo, and the lenses are separated by a distance 
d — /( 0) -f /( ir„), where W v is the v-aperture of the lens. Thus, the image i(xi,yi) of an input object 
o(x 0 } y 0 ) produced by the ALFE phase <f>D(u,v) is 


*(*<> w) = — ex P \J 


-1 

\ 2 z 


, 4~ x) 

’ A 


ex p + 3/ 2 )] J J exp[j(fo>(-u,-7j)] (9) 

y o(x 0 ,y 0 ) exp[j4> D (x 0l y 0 )} exp - «) 2 + (y„ - v) 2 ]| d.x 0 dy B 


1 2 * . . 

x exp — (tiZj + I’J/t ) 


du dv, 


where Fourier transformation is used to represent Frau nhofer diffraction to a distance z in the far-field. For 
a unit amplitude plane wave input, Fig. 7 represents simulated image reconstructions generated by Eq. (9) 
for a continuous phase zone plate lens and lenses designed using DS with phase quantization levels of 2, 4, 
and 8. N o cylindrical lens was used and the diffractive lens has a focal change from 240 mm to 384 min 
over a 3.2 mm aperture (// 75 to //120). The smallest feature is 25 Use of a cylindrical lens only 
changes the magnification and not the diffractive lens behavior. The magnification changes from 1.6 to 

0.625 over the extent of the aperture. 

It is apparent from Fig. 7 that a binary-phase DS conical lens is incapable of performing in the 
telescopic system. Diffractive artifacts are apparent even in the zone-plate image and it is interesting to 
note that more of the image is filled in as the number of phase quantization levels increases. The effect of 
variable magnification is also evident by the varying levels of image intensity; as the magnification 
increases, the intensity level decreases. 

To account for the undesirable effects of deterministically designed lenses, it is possible to design 
P(u,v) using iterative techniques to create a desired output image for a given input object. Because, from 
Eq. (9), the desired image q(xi,yi) can be determined foT a given input o(x 01 y 0 ) 1 it is possible to use 
iterative techniques, such as simulated annealing [2] andlterative Fourier transform algorithms [3], to 
determine such, that the error between z( , t/» ) and is a minimum. We will be 

investigating this technique in the future. 

4. Conclusion 


We have considered the design of a hybrid diffractive-refractive element that is capable of generating 
a focal line in space that has an arbitrarily specified shape. The element consists of a refractive cylindrical 
lens to provide optical power and a diffractive element for shaping the focal line. Although standard design 
and fabrication techniques have been applied to generate a diffractive conical lens, issues such as 
performance characterization and design for “improved” performance remain to be addressed more 
critically. Of utmost importance is data management to reduce design complexity, especially when 
placement accuracy is used as part of the design. These issues will be addressed in the future. 


References 


1. G. J. Swanson, MIT Lincoln Laboratory Technical Report 854 > 14 August 1989. 

2. W. II. Welch, J. E. Morris, and M. R. Feldman, “Iterative discrete on-axis encoding of radially 
symmetric computer generated holograms,” submitted to Appl. Opt. 

3. F. Wyrowski and O. Bryngdahl, Rep. Prog. Phys., 1481-1571 (1991). 


102 


mi mi ii i i i i i i i ii I ii 




(a) (b) (c) (d) 

Figure 2. (a) Binary mask designed using DS method and used to fabricate 
binary-phase conical lens. Lens reconstructions at (b) ~220 mm, (c) 
intermediate distance, and (d) ~350 mm. 



Figure 3. Calculated diffraction efficiency of DS designed binary-phase conical lens. 


103 


■ I'Jl'iU l W dT 








mask 1 


mask 2 


phase 



□ indicates single mask exposure 



(a) (b) 

Figure 5. Conical lenses designed using (a) direct sampling and (b) analytic 
quantization. 



Figure 6. Gain in diffraction efficiency for AQ-designed lens of Fig. 5(b) over 
DS-designed lens of Fig. 5(a). 


104 


I III! II Mil Hill I mm in I II i ilm .1 im in hh « i tun iii'timuMnAtyMii 









Figure 7. Simulated output image of phase-only conical lens illuminated by unit 
amplitude plane wave assuming: (a) continuous phase, (b) 2 phase levels, (c) 4 
phase levels, and (d) 8 phase levels. 


105